Animation: six point vortices moving through the velocity field they induce, drawn as streamlines over a map of flow speed, with filled markers rotating counter-clockwise and open markers clockwise. — The 2D Ising model

Spins $s_i = \pm 1$ on an $n \times n$ lattice with periodic boundaries, coupled ferromagnetically:
$E = -J \sum_{\langle i,j \rangle} s_i s_j, \qquad J = 1$
A Metropolis sweep attempts $n^2$ spin flips; a flip that raises the energy by $\Delta E$ is accepted with probability $e^{-\Delta E / k_B T}$ . Below $T_c = 2J / k_B \ln(1+\sqrt{2}) \approx 2.269$ the lattice magnetizes. Lemma compiled the sweep with Cython and checked it bit-for-bit against a pure-Python reference before running the physics.

Point vortices in a 2D flow

$N$ point vortices in an unbounded inviscid flow, each advected by the velocity every other vortex induces:
$u_i = -\frac{1}{2\pi}\sum_{j\neq i}\Gamma_j \frac{y_i-y_j}{r_{ij}^2}, \qquad v_i = \frac{1}{2\pi}\sum_{j\neq i}\Gamma_j \frac{x_i-x_j}{r_{ij}^2}$
A Hamiltonian system since Kirchhoff (1876). Lemma integrated it with RK4 and checked the integrator against the conserved $H$ , $Q = \sum_i \Gamma_i x_i$ , $P = \sum_i \Gamma_i y_i$ , and $I = \sum_i \Gamma_i (x_i^2 + y_i^2)$ , and against the two-vortex cases it could solve by hand.

The six trajectories plotted in three dimensions with time on the vertical axis: helices where pairs orbit each other, near-straight lines where pairs run off together. — The 2D Ising model

Spins $s_i = \pm 1$ on an $n \times n$ lattice with periodic boundaries, coupled ferromagnetically:
$E = -J \sum_{\langle i,j \rangle} s_i s_j, \qquad J = 1$
A Metropolis sweep attempts $n^2$ spin flips; a flip that raises the energy by $\Delta E$ is accepted with probability $e^{-\Delta E / k_B T}$ . Below $T_c = 2J / k_B \ln(1+\sqrt{2}) \approx 2.269$ the lattice magnetizes. Lemma compiled the sweep with Cython and checked it bit-for-bit against a pure-Python reference before running the physics.

Point vortices in a 2D flow

$N$ point vortices in an unbounded inviscid flow, each advected by the velocity every other vortex induces:
$u_i = -\frac{1}{2\pi}\sum_{j\neq i}\Gamma_j \frac{y_i-y_j}{r_{ij}^2}, \qquad v_i = \frac{1}{2\pi}\sum_{j\neq i}\Gamma_j \frac{x_i-x_j}{r_{ij}^2}$
A Hamiltonian system since Kirchhoff (1876). Lemma integrated it with RK4 and checked the integrator against the conserved $H$ , $Q = \sum_i \Gamma_i x_i$ , $P = \sum_i \Gamma_i y_i$ , and $I = \sum_i \Gamma_i (x_i^2 + y_i^2)$ , and against the two-vortex cases it could solve by hand.

All of the science, less of the pain.

Bring Lemma the work that takes a week when you only have a morning. You direct the approach and approve the plan. Lemma does the work: managed, verified, and reported back.

Approved specifications

You set the direction: refine the plan, compare approaches, and set the specifications each step must meet. Nothing runs until you approve, and when a step fails, Lemma asks again.
Managed environment

Lemma runs the solvers, simulators and optimizers of your field, and installs anything else on request. No setup, no version conflicts.
Verification pipeline

Lemma checks results as they are produced for consistency, coherence, and agreement with sources, and reports when an output doesn't yet meet your specifications.
Interactive reports

Lemma reports back in interactive notebooks, where code and plots annotate the outcome. Portable anywhere.

What you can do with Lemma

Every example below opens in Lemma with the prompt ready to run. You will need a free account.

Explore ideas

An idea shouldn’t need an afternoon of setup just to look at it. Ask, from a derivation to a design trade-off, and get back something that runs.

Understand

Guided exploration

A topic built up in a live notebook, at your level, one runnable step at a time.
Try it in Lemma “Walk me through the quantum Hall effect in graphene, building the Landau-level spectrum step by step.”
Derivation

Results derived from first principles, alternative routes compared.
Try it in Lemma “Derive the Berry curvature of a gapped Dirac cone, and show the Kubo-formula route as a cross-check.”
Problem solving

A concrete answer to a concrete question, worked end to end.
Try it in Lemma “How many helium balloons would it take to lift a laptop?”

Interrogate

Parameter exploration

Sweeps, scaling laws, regime boundaries.
Try it in Lemma “Simulate the Beddington–DeAngelis predator-prey model and draw a phase diagram over its parameters.”
Scenario comparison

Designs, models, or platforms side by side, with the trade-offs shown.
Try it in Lemma “Compare a Si and a SiN grating coupler at 1550 nm on efficiency and bandwidth.”
Experiment planning

Measurement plans with error budgets, before spending lab time.
Try it in Lemma “Plan a magnetotransport run to resolve the fractional states, with an error budget for the field sweep.”

Build & simulate

Build the simulation you’ve been dreaming of. No environment setup, no boilerplate, and no solver docs in the way. Describe it, and watch every step run.

Computational methods

Model building

Describe a device or a phenomenon and get a working model of it, built and run.
Try it in Lemma “Build a model of a Gunn diode and show its negative differential resistance.”
Model comparison

Competing models ranked by information criteria.
Try it in Lemma “Simulate two-carrier magnetoresistance data, then rank single- vs two-carrier Drude fits by AIC.”
Paper reproduction

Rebuild a paper as a live notebook reproducing its results.
Try it in Lemma “Rebuild Castilla et al.'s graphene THz detector paper (arXiv:1905.01881) and confirm the key figure.”

Mathematics

Equation checking

Dimensional analysis and symbolic verification of derivations.
Try it in Lemma “Derive the damped-pendulum frequency, checking units at each step, and flag any sign errors.”
Formal proof

Machine-checked proofs in Lean 4 with mathlib, for the ultimate layer of rigour.
Try it in Lemma “Prove in Lean that the energy functional E[ψ] = ∫|∇ψ|² + V(x)|ψ|² dx is convex.”

Optimization

Numerical optimization

Constrained optimization, ODE/DAE simulation, optimal control.
Try it in Lemma “Find the layer thicknesses that minimize reflection at 1550 nm under a ±20 nm fab tolerance.”
Inverse design

Gradient-based design through differentiable simulation.
Try it in Lemma “Inverse-design a compact waveguide taper reaching >99% transmission with differentiable FDTD.”

Solvers

Foundations

Review & draft

Days of checking a paper’s math by hand, done in minutes. Verify the papers you read, draft the ones you write.

Read

Document analysis

Analyze equations, figures, tables, and claims extracted from any given document.
Try it in Lemma “Pull every equation, figure, and claim out of Kim et al.'s photonics review (arXiv:1911.12756).”
Validation report

A structured report on a paper's references, equations and claims, scoped to whichever you want checked.
Try it in Lemma “Generate a validation report on the equations and references of arXiv:1610.02365.”

Write

Manuscripts

LaTeX drafted from your own results, bibliography included, compiled to PDF.
Try it in Lemma “Simulate a Lindblad master equation, then draft a LaTeX methods section from the results.”
Grant proposals

Drafted against specific calls, grounded in your results.
Try it in Lemma “Draft an EIC Pathfinder concept for a graphene THz detector, grounded in a feasibility estimate.”
Slides

Talk-ready decks, built from a paper or from your own results.
Try it in Lemma “Build a .pptx deck reviewing arXiv:1610.02365 (photonic deep learning).”

Tools

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All of the science, less of the pain.

Approved specifications

Managed environment

Verification pipeline

Interactive reports

What you can do with Lemma

Explore ideas

Guided exploration

Derivation

Problem solving

Parameter exploration

Scenario comparison

Experiment planning

Build & simulate

Model building

Model comparison

Paper reproduction

Equation checking

Formal proof

Numerical optimization

Inverse design

Review & draft

Document analysis

Validation report

Manuscripts

Grant proposals

Slides

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